Learn Spanish (Latin America): Ella sigue superando esos obstáculos.

The combination of seguir (to continue or to follow) and a gerund (a verb ending in -ando or -iendo) is used to emphasize that an action is ongoing. So, sigue superando literally means 'continues overcoming', which we naturally translate as 'keeps overcoming' in English.

I always mix up esos, estos, and aquellos. Why use esos here?

Esos means 'those' (for masculine plural nouns like obstáculos) when the things are at a medium distance, or just generally 'those things we just mentioned'. If they were right here with us, we would use estos (these). If they were far away in time or space, we would use aquellos (those over there).

Why does the word obstáculos have an accent mark on the 'a'?

In Spanish, words naturally stress the second-to-last syllable if they end in a vowel, the letter 'n', or the letter 's'. Without the accent, we would emphasize the 'cu' syllable. Because the spoken stress actually falls on the third-to-last syllable (the 'ta'), we must write an accent mark to break the standard rule.

What is the base verb for superando, and what else can it mean?

The base verb is superar. Aside from 'to overcome' a problem or obstacle, it can also mean 'to surpass' a record or expectation, or 'to get over' a difficult situation.